Using Weighted Partial Least Squares to predict Brain structures

1
Using Weighted Partial Least Squares to predict
Brain structures

• Anil Rao
• Department of Computing
• Imperial College London

2
Modelling inter-structure variation

• Want to explicitly model variation between 2
  structures X and Y
• Joint P.C.A. would include intra-structure
  variation
• Canonical Correlation Analysis
• Identifies most correlated modes of X and Y and
  gives corresponding correlation coefficient
• Coordinates of highly correlated modes can then
  be predicted for Y, given corresponding
  coordinates for X
• Poorly correlated modes cannot be used for
  prediction, so whole of Y cannot be predicted
  from X
• Partial Least Squares Regression (PLSR)
• Predicts whole of Y in one step

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Partial Least Squares Regression (1)

• Idea
• X (structure 1) is set of predictor signals x, Y
  (structure 2) is set of response signals y
• Produce a linear model
• are mean-centred, normalized by
  standard deviation
• Directions in X sought that describe most
  variation in Y
• C then used to predict instance of Y from that of
  X
• Implementation
• NIPALS algorithm
• Iterative technique for calculating C
• Inputs N datasets of X (dimension p),Y
  (dimension q)
• Outputs C, means and standard deviations

4
Partial Least Squares Regression (1)

• Idea
• X (structure 1) is set of predictor signals x, Y
  (structure 2) is set of response signals y
• Produce a linear model
• are mean-centred, normalized by standard
  deviation over X,Y
• Directions in X sought that describe most
  variation in Y
• C then used to predict instance of y from that of
  x

5
Partial Least Squares Regression (2)

• Implementation
• NIPALS algorithm
• Iterative technique for calculating C, uses
  covariance matrices
• Inputs N datasets of X (dimension p),Y
  (dimension q)
• Outputs C (pxq), means and standard
  deviations of X,Y

6
Application to brain data

• PLSR used to model relationship between pairs of
  structures over 37 data sets
• Separate PCA of each structure used to reduce
  dimensionality, retained all modes
• PLSR performed on reduced data, predictions then
  transformed back to original basis
• Leave one out tests performed
• Errors between predicted shape and actual shape
  calculated
• Compared to distances between predicted shape and
  mean of that shape

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Correlation image (from CCA)
Structure j
Structure i
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Results left/right thalamus
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Results left/right thalamus

• Prediction of right thalamus shape better than
  mean in 32/37 cases
• Biggest improvement 7.07 to 1.81
• Worst case 1.14 to 1.72
• Cases that are worse associated with poor PCA
  fitting of unseen left thalamus
• Average prediction error smaller than mean
• Average error of mean2.60
• Average error of prediction1.23

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Results left lat ventricle /right putamen
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Results left lateral ventricle/right putamen

• Prediction of right putamen shape better than
  mean in 18/37 cases
• Biggest improvement 7.65 to 4.94
• Worst case 8.38 to 10.17
• Average prediction error similar to mean
• Average error of mean2.36
• Average error of prediction2.29
• Is poor performance related to low canonical
  correlation coefficient?
• PLSR is linear technique, low cca measures imply
  variation is not linear

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Results right pallidum /right putamen
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Results right pallidum/right putamen

• Prediction of right putamen shape better than
  mean in 36/37 cases
• Biggest improvement 7.65 to 0.88
• Worst case 0.96 to 1.04
• Average prediction error smaller than mean
• Average error of mean2.36
• Average error of prediction0.79
• Result better than left thal/right thal
• Average PCA fitting error much smaller for unseen
  predictor structure left thal0.75,right
  pall0.23

14
Canonical correlations of brain structures

• Pairwise CCA analysis performed of 17 brain
  structures over 37 data sets
• Separate PCA of each structure used to reduce
  dimensionality first
• First 17 modes (gt95) of each structure retained
• CCA then performed on each pair of structures
• Average correlations (0-1) between structures
  calculated

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Results- Most Correlated Structures
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Results- Most Correlated Structures